Consider the continuous-time system given by the state equations x(t) = =| X(t) + -1.5 1 1 utt) y₁ (t) = [10]x(t) (1) y₂

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Consider The Continuous Time System Given By The State Equations X T X T 1 5 1 1 Utt Y T 10 X T 1 Y 1 (32.9 KiB) Viewed 37 times

Consider the continuous-time system given by the state equations x(t) = =| X(t) + -1.5 1 1 utt) y₁ (t) = [10]x(t) (1) y₂(t) = [0 -1]x(t). (a) [2 marks] Find the system transfer functions from u to y, and from u to y₂. (b) [10 marks] Design a state-feedback control law with integral action to achieve robust tracking of step references r in y₂(t), that is, u(t)= -Kx(t) - K₂z(t) ż(t) = r - y₂(t). Find the matrices K and K₂ to place all the closed-loop eigenvalues at -2. Would it be possible to achieve robust tracking of constant references also in y₁ (t)? Justify your answer. (c) [6 marks] Write the equations of an observer for the system (1) using only y₁ (t) as mea- surement, and design the observer gains in order to obtain an estimate X(t) of the state with observer eigenvalues at -10.